Tangent function is defined as the ratio of the side perpendiculardivided by the adjacent.
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If we started from A và moves in anticlockwise direction then at the points A, B, A", B" và A, the arc length travelled are 0, (fracπ2), π, (frac3π2), và 2π.
tan θ = (fracPMOM)
Now, chảy θ = 0
⇒ (fracPMOM) = 0
⇒ PM = 0.
So when will the tangent be equal lớn zero?
Clearly, if PM = 0 then the final arm OP of the angle θcoincides with OX or OX".
Similarly, the final arm OPcoincides with OX or OX" when θ = π, 2π, 3π, 4π, ……….. , - π, -2π, -3π,-4π, ……….. I.e. When θ an integral multiples of π i.e., when θ = nπ where n ∈Z (i.e., n = 0, ± 1, ± 2, ± 3,…….)
Hence, θ = nπ, n ∈Z is the general solution of the given equation chảy θ = 0
1. Find the general solution of the equation rã 2x = 0
Solution:
tan 2x = 0
⇒ 2x = nπ, where, n = 0, ± 1, ± 2, ± 3, …….
⇒ x = (fracnπ2), where, n = 0, ± 1, ± 2, ± 3, …….
Therefore, the general solution of the trigonometric equation tung 2x = 0 is x = (fracnπ2), where, n = 0, ± 1, ± 2, ± 3, …….
2. Find the general solution of the equation rã (fracx2) = 0
Solution:
tan (fracx2) = 0
⇒ (fracx2) = nπ, where, n = 0, ± 1, ± 2, ± 3, …….
⇒ x = 2nπ, where, n = 0, ± 1, ± 2, ± 3, …….
Therefore, the general solution of the trigonometric equation rã (fracx2) = 0 is x = 2nπ, where, n = 0, ± 1, ± 2, ± 3, …….
3. What is the general solution of the equation tung x + chảy 2x + tung 3x = tan x rã 2x rã 3x?
Solution:
tan x + rã 2x + tan 3x = chảy x tan 2x chảy 3x
⇒ rã x + tung 2x = - tung 3x + rã x rã 2x tung 3x
⇒ tan x + tung 2x = - tung 3x(1 - tung x chảy 2x)
⇒ (fractan x + chảy 2x1 - tung x chảy 2x) = - tung 3x
⇒ chảy (x + 2x) = - chảy 3x
⇒ tung 3x = - chảy 3x
⇒ 2 rã 3x = 0
⇒ chảy 3x = 0
⇒ 3x = nπ, where n = 0, ± 1, ± 2, ± 3,…….
x = (fracnπ3), where n = 0, ± 1, ± 2, ± 3,…….
Therefore, the general solution of the trigonometric equation chảy x + chảy 2x + chảy 3x = rã x rã 2x tan 3x is x = (fracnπ3), where n = 0, ± 1, ± 2, ± 3,…….
4. Find the general solution of the equation chảy (frac3x4) = 0
Solution:
tan (frac3x4) = 0
⇒ (frac3x4) = nπ, where, n = 0, ± 1, ± 2, ± 3, …….
⇒ x = (frac4nπ3), where, n = 0, ± 1, ± 2, ± 3, …….
Therefore, the general solution of the trigonometric equation tan (frac3x4) = 0 is x = (frac4nπ3), where, n = 0, ± 1, ± 2, ± 3, …….
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● Trigonometric Equations
11 and 12 Grade Math
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